The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise

243 Pages·2004·1.91 MB·English

by  Mary Tiles

Copyright
Copyright © 1989 by Mary Tiles All rights reserved.

Bibliographical Note This Dover edition, first published in 2004, is an
unabridged republication of the work originally published in 1989 by Basil
Blackwell Ltd., Oxford, United Kingdom, and Basil Blackwell Inc., Cambridge,
Massachusetts.

Library of Congress Cataloging-in-Publication Data Tiles, Mary.
The philosophy of set theory : an historical introduction to Cantor’s paradise /
Mary Tiles.
p. cm.
Originally published: Oxford, UK : Cambridge, Mass., U.S.A. : Blackwell,
1989.
Includes bibliographical references and index.
9780486138558
1. Mathematics—Philosophy. 2. Set theory. I. Title.
QA8.4.T54 2004
511.3’22—dc22
2004043941

Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Like the enigma of time for Augustine, the enigma of the continuum arises
because language misleads us into applying to it a picture that doesn’t fit. Set
theory preserves the inappropriate picture of something discontinuous, but
makes statements about it that contradict the picture, under the impression that
it is breaking with prejudices; whereas what should really have been done is to
point out that the picture just doesn’t fit, that it certainly can’t be stretched
without being torn, and that instead of it one can use a new picture in certain
respects similar to the old one.
Wittgenstein, 1974, p. 471
For Jim
in memory of theses
‘not dead, but sleeping’
Table of Contents
Title Page
Copyright Page
Epigraph
Dedication
Preface
Introduction: Invention or Discovery?
1 - The Finite Universe
2 - Classes and Aristotelian Logic
3 - Permutations, Combinations and Infinite Cardinalities
4 - Numbering the Continuum
5 - Cantor’s Transfinite Paradise
6 - Axiomatic Set Theory
7 - Logical Objects and Logical Types
8 - Independence Results and the Universe of Sets
9 - Mathematical Structure – Construct and Reality
Further Reading
Bibliography
Glossary of Symbols
Index
Preface
This book is the result of a number of attempts to teach undergraduate classes
and seminars in the philosophy of mathematics, sometimes to a mixture of
mathematicians and philosophers, sometimes just to interested philosophers. I
have found it difficult to recommend reading that was neither too
mathematically technical for the philosophers nor too philosophically technical
for the mathematicians. I confess to failing wholly to resolve that problem,
although that was the aim. The hope is nevertheless that both philosophers with
only a very basic grounding in mathematics and mathematicians who have taken
only an introductory course in philosophy may find something of interest by
differentially skipping over those parts which are either too technical or too
familiar. I would like in particular to persuade philosophers that the philosophy
of mathematics is not an isolated speciality but is inseparably intertwined with
what are standardly regarded as mainstream philosophical issues.
Thanks are due to Dr Jim Brown of Toronto University for encouraging the
project and for test running the draft version, and to his students for their
comments and corrections; also to Hal Switkay for pointing out some
mathematical errors. I would like to thank also Swarthmore students David
Ravinsky and Russell Marcus for their suggestions and for letting me try out
material on them. Above all thanks to Jim Tiles for all those innumerable forms
of support without which the book would never have been completed.
Introduction: Invention or
Discovery?
Did Cantor discover the rich and strange world of transfinite sets (which Hilbert
was to call Cantor’s Paradise) or did he (with a little help from his friends) create
it? Are set theorists now discovering more about the universe to which Cantor
showed them the way, or are they continuing the creative process? Perhaps they
are wandering in a wonderland which is no more understandable and no more
substantial than that in which Alice, in Lewis Carroll’s Alice in Wonderland,
found herself.
One way of approaching these questions is to think about them in relation to a
question to which Cantor devoted much time in his later years: ‘How many
points are there in a line?’ Cantor thought he knew the answer. ‘There are
and = ’ ( , aleph, is the first letter of the Hebrew alphabet.) Here is the
1 0
first infinite cardinal number. It is the number of the set of natural numbers, {0,
1, 2, 3, . . .}, irrespective of the order in which they are counted. is to be
understood by analogy with 22, 23, 24, etc. and is the next largest infinite
1
cardinal number after . = , which has come to be known as Cantor’s
0 1
continuum hypothesis, thus says that the number of points on a line is the second
infinite cardinal number: there are none in between and Notice that we
0
would not reach this conclusion by generalizing from 2n. For example 8 (= 23) is
not the next number after 3, and in general 2n is not the next number after n. But
infinite numbers are strange things and we should not expect them to behave in
all respects like finite numbers.
Cantor’s problem was that, although convinced of the correctness of his
hypothesis, he never succeeded in proving it. Moreover, subsequent work in set
theory has not resolved the question. We may know a lot more about the
problem, but we also know that Cantor’s continuum hypothesis cannot be proved
from the standardly accepted axioms of set theory. What then, should be our
attitude towards his hypothesis and toward possible answers to the question
concerning the number of points in a line? This all depends on the answer given
concerning the number of points in a line? This all depends on the answer given
to the questions with which we started.
Suppose Cantor did discover a new realm, a realm which has now been more
extensively explored and systematically mapped since the axiomatization of set
theory. Then his hypothesis, being a hypothesis about things in this realm, must
be either correct or incorrect even though our present axioms do not characterize
this realm sufficiently precisely for a proof to be given which would enable us to
determine which is the case. Only by discovering new axioms, new fundamental
truths about the set theoretic universe, will it be possible to give proofs which
would settle the matter. But how are these new axioms to be discovered? How
do we gain access to the uncharted areas of this realm? These would be
questions which could not be avoided.
On the other hand, suppose that Cantor created the realm of infinite sets and
infinite numbers in the way that Lewis Carroll created Wonderland. Although
creations of this sort can be discussed, analysed and schematized by others (we
might try constructing a map of Wonderland), they are nevertheless such that
there will be some questions about them which simply have no answers because
the creator did not supply enough information to provide an answer and there is
no other source of information. The question ‘What was the diameter of the top
of the Madhatter’s hat?’ has no answer, for Alice in Wonderland neither includes
this directly in its description nor provides details of other dimensions from
which an answer could be deduced. Given that this is so, we could consistently
add to the story by filling in this detail (in the way that illustrators have filled in
the price of the hat – Tenniel gives it a price tag of 10/6d, whereas that in
Rackham’s illustration is 8/4d) and we might do so in different ways, so
continuing the creative process. Of course we might also argue that dimensions
in Wonderland are peculiarly problematic; given Alice’s tendency to grow and
shrink we would have to specify the frame of reference carefully. It seems that
the continuum hypothesis is a similarly unanswerable question about infinite
numbers. There are several different ways in which we can add to the basic set
theoretic axioms, so giving rise to different extended set theories, each of which
would be seen as a filling out of the original. There are some constraints on what
number is assigned to the continuum, but there is also a very considerable degree
of freedom. If mathematics is a free creative activity, constrained only by
demands of consistency, absence of contradiction, then all of these alternative
extended set theories have an equal claim to mathematical legitimacy.
It might, however, be argued that the continuum hypothesis was not originally
asked as a question about a self-contained realm of infinite sets created by